August  2017, 11(4): 601-621. doi: 10.3934/ipi.2017028

Landmark-guided elastic shape analysis of human character motions

1. 

Florida State University, Department of Mathematics, 208 Love Building, 1017 Academic Way, Tallahassee, FL 32306-4510, USA

2. 

Department of Mathematical Sciences, NTNU -Norwegian University of Science and Technology, 7491 Trondheim, Norway

* Corresponding author: Martin Bauer

Received  December 2014 Revised  March 2017 Published  June 2017

Fund Project: MB was supported by FWF project P24625.

Motions of virtual characters in movies or video games are typically generated by recording actors using motion capturing methods. Animations generated this way often need postprocessing, such as improving the periodicity of cyclic animations or generating entirely new motions by interpolation of existing ones. Furthermore, search and classification of recorded motions becomes more and more important as the amount of recorded motion data grows.

In this paper, we will apply methods from shape analysis to the processing of animations. More precisely, we will use the by now classical elastic metric model used in shape matching, and extend it by incorporating additional inexact feature point information, which leads to an improved temporal alignment of different animations.

Citation: Martin Bauer, Markus Eslitzbichler, Markus Grasmair. Landmark-guided elastic shape analysis of human character motions. Inverse Problems and Imaging, 2017, 11 (4) : 601-621. doi: 10.3934/ipi.2017028
References:
[1]

M. F. AbdelkaderW. Abd-AlmageedA. Srivastava and R. Chellappa, Silhouette-based gesture and action recognition via modeling trajectories on Riemannian shape manifolds, Computer Vision and Image Understanding, 115 (2011), 439-455.  doi: 10.1016/j.cviu.2010.10.006.

[2]

D. AlekseevskyA. KrieglM. Losik and P. W. Michor, The Riemannian geometry of orbit spaces--the metric, geodesics, and integrable systems, Publ. Math. Debrecen, 62 (2003), 247-276. 

[3]

M. Bauer, M. Bruveris, P. Harms and J. M. Andersen, Curve matching with applications in medical imaging, in 5th MICCAI Workshop on Mathematical Foundations of Computational Anatomy, 2015.

[4]

M. Bauer, M. Bruveris, P. Harms and J. M. Andersen, Second order elastic metrics on the shape space of curves in, Proceedings of the 1st International Workshop on DIFFerential Geometry in Computer Vision for Analysis of Shapes, Images and Trajectories (DIFF-CV 2015) (eds. H. Drira, S. Kurtek and P. Turaga), BMVA Press, 2015, pp 9. doi: 10.5244/C.29.DIFFCV.9.

[5]

M. BauerM. BruverisP. Harms and J. M. Andersen, A numerical framework for sobolev metrics on the space of curves, SIAM J. Imaging Sci., 10 (2017), 47-73.  doi: 10.1137/16M1066282.

[6]

M. BauerM. BruverisS. Marsland and P. W. Michor, Constructing reparameterization invariant metrics on spaces of plane curves, Differential Geom. Appl., 34 (2014), 139-165.  doi: 10.1016/j.difgeo.2014.04.008.

[7]

M. BauerM. Bruveris and P. W. Michor, Overview of the geometries of shape spaces and diffeomorphism groups, J. Math. Imaging Vision, 50 (2014), 60-97.  doi: 10.1007/s10851-013-0490-z.

[8]

M. BauerM. Bruveris and P. W. Michor, R-transforms for Sobolev H2-metrics on spaces of plane curves, Geometry, Imaging and Computing, 1 (2014), 1-56.  doi: 10.4310/GIC.2014.v1.n1.a1.

[9]

M. Bauer and P. Harms, Metrics on spaces of surfaces where horizontality equals normality, Differential Geom. Appl., 39 (2015), 166–183, http://arxiv.org/abs/1403.1436. doi: 10.1016/j.difgeo.2014.12.008.

[10]

A. Bruderlin and L. Williams, Motion signal processing, Proceedings of the 22nd annual conference on Computer graphics and interactive techniques, ACM, (1995), 97-104.  doi: 10.1145/218380.218421.

[11]

M. Bruveris, P. W. Michor and D. Mumford, Geodesic completeness for sobolev metrics on the space of immersed plane curves Forum of Mathematics, Sigma, 2 (2014), e19, 38 pp. doi: 10.1017/fms.2014.19.

[12]

Carnegie-Mellon, Carnegie-mellon mocap database, 2003.

[13]

E. CelledoniM. Eslitzbichler and A. Schmeding, Shape analysis on Lie groups with application in computer animation, Journal of Geometric Mechanics, 8 (2016), 273-304.  doi: 10.3934/jgm.2016008.

[14]

M. Eslitzbichler, Modelling character motions on infinite-dimensional manifolds, The Visual Computer, 31 (2015), 1179-1190.  doi: 10.1007/s00371-014-1001-y.

[15]

M. FuchsB. JüttlerO. Scherzer and H. Yang, Shape metrics based on elastic deformations, J. Math. Imaging Vision, 35 (2009), 86-102.  doi: 10.1007/s10851-009-0156-z.

[16]

L. Kovar and M. Gleicher, Flexible automatic motion blending with registration curves, in Proceedings of the 2003 ACM SIGGRAPH/Eurographics Symposium on Computer Animation, SCA '03, Eurographics Association, Aire-la-Ville, Switzerland, 2003,214–224.

[17]

L. Kovar and M. Gleicher, Automated extraction and parameterization of motions in large data sets, ACM Transactions on Graphics (TOG), 23 (2004), 559-568.  doi: 10.1145/1186562.1015760.

[18]

S. KurtekA. SrivastavaE. Klassen and H. Laga, Landmark-guided elastic shape analysis of spherically-parameterized surfaces, Computer Graphics Forum, 32 (2013), 429-438.  doi: 10.1111/cgf.12063.

[19]

H. LagaS. KurtekA. Srivastava and S. J. Miklavcic, Landmark-free statistical analysis of the shape of plant leaves, J. Theoret. Biol., 363 (2014), 41-52.  doi: 10.1016/j.jtbi.2014.07.036.

[20]

W. Liu, A Riemannian Framework For Annotated Curves Analysis, PhD thesis, The Florida State University, 2011.

[21]

A. MennucciA. Yezzi and G. Sundaramoorthi, Properties of Sobolev-type metrics in the space of curves, Interfaces Free Bound, 10 (2008), 423-445.  doi: 10.4171/IFB/196.

[22]

P. W. Michor and D. Mumford, An overview of the Riemannian metrics on spaces of curves using the {H}amiltonian approach, Appl. Comput. Harmon. Anal., 23 (2007), 74-113.  doi: 10.1016/j.acha.2006.07.004.

[23]

W. Mio and A. Srivastava, Elastic-string models for representation and analysis of planar shapes, In Computer Vision and Pattern Recognition, 2004. CVPR 2004. Proceedings of the 2004 IEEE Computer Society Conference on, volume 2, pages (2004), 10–15. doi: 10.1109/CVPR.2004.1315138.

[24]

W. MioA. Srivastava and S. Joshi, On shape of plane elastic curves, Int. J. Comput. Vision, 73 (2007), 307-324.  doi: 10.1007/s11263-006-9968-0.

[25]

T. Pejsa and I. Pandzic, State of the art in example-based motion synthesis for virtual characters in interactive applications, Computer Graphics Forum, 29 (2010), 202-226.  doi: 10.1111/j.1467-8659.2009.01591.x.

[26]

C. SamirP.-A. AbsilA. Srivastava and E. Klassen, A gradient-descent method for curve fitting on Riemannian manifolds, Found. Comput. Math., 12 (2012), 49-73.  doi: 10.1007/s10208-011-9091-7.

[27]

J. Shah, $H^0$-$ type Riemannian metrics on the space of planar curves, Quart. Appl. Math., 66 (2008), 123-137.  doi: 10.1090/S0033-569X-07-01084-4.

[28]

J. Shah, An $H^2$ Riemannian metric on the space of planar curves modulo similitudes, Adv. in Appl. Math., 51 (2013), 483-506.  doi: 10.1016/j.aam.2013.06.003.

[29]

E. Sharon and D. Mumford, 2D-shape analysis using conformal mapping, International Journal of Computer Vision, 70 (2004), 55-75.  doi: 10.1109/CVPR.2004.1315185.

[30]

A. SrivastavaE. KlassenS. Joshi and I. Jermyn, Shape analysis of elastic curves in euclidean spaces, Pattern Analysis and Machine Intelligence, IEEE Transactions on, 33 (2011), 1415-1428.  doi: 10.1109/TPAMI.2010.184.

[31]

J. SuS. KurtekE. Klassen and A. Srivastava, Statistical analysis of trajectories on Riemannian manifolds: bird migration, hurricane tracking and video surveillance, Ann. Appl. Stat., 8 (2014), 530-552.  doi: 10.1214/13-AOAS701.

[32]

Q. XieS. KurtekE. Klassen and A. Srivastava, Analysis of AneuRisk65 data: Elastic shape registration of curves, Electron. J. Stat., 8 (2014), 1920-1929.  doi: 10.1214/14-EJS938D.

[33]

L. Younes, Computable elastic distances between shapes, SIAM J. Appl. Math., 58 (1998), 565-586 (electronic).  doi: 10.1137/S0036139995287685.

[34]

L. YounesP. W. MichorJ. Shah and D. Mumford, A metric on shape space with explicit geodesics, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 19 (2008), 25-57.  doi: 10.4171/RLM/506.

show all references

References:
[1]

M. F. AbdelkaderW. Abd-AlmageedA. Srivastava and R. Chellappa, Silhouette-based gesture and action recognition via modeling trajectories on Riemannian shape manifolds, Computer Vision and Image Understanding, 115 (2011), 439-455.  doi: 10.1016/j.cviu.2010.10.006.

[2]

D. AlekseevskyA. KrieglM. Losik and P. W. Michor, The Riemannian geometry of orbit spaces--the metric, geodesics, and integrable systems, Publ. Math. Debrecen, 62 (2003), 247-276. 

[3]

M. Bauer, M. Bruveris, P. Harms and J. M. Andersen, Curve matching with applications in medical imaging, in 5th MICCAI Workshop on Mathematical Foundations of Computational Anatomy, 2015.

[4]

M. Bauer, M. Bruveris, P. Harms and J. M. Andersen, Second order elastic metrics on the shape space of curves in, Proceedings of the 1st International Workshop on DIFFerential Geometry in Computer Vision for Analysis of Shapes, Images and Trajectories (DIFF-CV 2015) (eds. H. Drira, S. Kurtek and P. Turaga), BMVA Press, 2015, pp 9. doi: 10.5244/C.29.DIFFCV.9.

[5]

M. BauerM. BruverisP. Harms and J. M. Andersen, A numerical framework for sobolev metrics on the space of curves, SIAM J. Imaging Sci., 10 (2017), 47-73.  doi: 10.1137/16M1066282.

[6]

M. BauerM. BruverisS. Marsland and P. W. Michor, Constructing reparameterization invariant metrics on spaces of plane curves, Differential Geom. Appl., 34 (2014), 139-165.  doi: 10.1016/j.difgeo.2014.04.008.

[7]

M. BauerM. Bruveris and P. W. Michor, Overview of the geometries of shape spaces and diffeomorphism groups, J. Math. Imaging Vision, 50 (2014), 60-97.  doi: 10.1007/s10851-013-0490-z.

[8]

M. BauerM. Bruveris and P. W. Michor, R-transforms for Sobolev H2-metrics on spaces of plane curves, Geometry, Imaging and Computing, 1 (2014), 1-56.  doi: 10.4310/GIC.2014.v1.n1.a1.

[9]

M. Bauer and P. Harms, Metrics on spaces of surfaces where horizontality equals normality, Differential Geom. Appl., 39 (2015), 166–183, http://arxiv.org/abs/1403.1436. doi: 10.1016/j.difgeo.2014.12.008.

[10]

A. Bruderlin and L. Williams, Motion signal processing, Proceedings of the 22nd annual conference on Computer graphics and interactive techniques, ACM, (1995), 97-104.  doi: 10.1145/218380.218421.

[11]

M. Bruveris, P. W. Michor and D. Mumford, Geodesic completeness for sobolev metrics on the space of immersed plane curves Forum of Mathematics, Sigma, 2 (2014), e19, 38 pp. doi: 10.1017/fms.2014.19.

[12]

Carnegie-Mellon, Carnegie-mellon mocap database, 2003.

[13]

E. CelledoniM. Eslitzbichler and A. Schmeding, Shape analysis on Lie groups with application in computer animation, Journal of Geometric Mechanics, 8 (2016), 273-304.  doi: 10.3934/jgm.2016008.

[14]

M. Eslitzbichler, Modelling character motions on infinite-dimensional manifolds, The Visual Computer, 31 (2015), 1179-1190.  doi: 10.1007/s00371-014-1001-y.

[15]

M. FuchsB. JüttlerO. Scherzer and H. Yang, Shape metrics based on elastic deformations, J. Math. Imaging Vision, 35 (2009), 86-102.  doi: 10.1007/s10851-009-0156-z.

[16]

L. Kovar and M. Gleicher, Flexible automatic motion blending with registration curves, in Proceedings of the 2003 ACM SIGGRAPH/Eurographics Symposium on Computer Animation, SCA '03, Eurographics Association, Aire-la-Ville, Switzerland, 2003,214–224.

[17]

L. Kovar and M. Gleicher, Automated extraction and parameterization of motions in large data sets, ACM Transactions on Graphics (TOG), 23 (2004), 559-568.  doi: 10.1145/1186562.1015760.

[18]

S. KurtekA. SrivastavaE. Klassen and H. Laga, Landmark-guided elastic shape analysis of spherically-parameterized surfaces, Computer Graphics Forum, 32 (2013), 429-438.  doi: 10.1111/cgf.12063.

[19]

H. LagaS. KurtekA. Srivastava and S. J. Miklavcic, Landmark-free statistical analysis of the shape of plant leaves, J. Theoret. Biol., 363 (2014), 41-52.  doi: 10.1016/j.jtbi.2014.07.036.

[20]

W. Liu, A Riemannian Framework For Annotated Curves Analysis, PhD thesis, The Florida State University, 2011.

[21]

A. MennucciA. Yezzi and G. Sundaramoorthi, Properties of Sobolev-type metrics in the space of curves, Interfaces Free Bound, 10 (2008), 423-445.  doi: 10.4171/IFB/196.

[22]

P. W. Michor and D. Mumford, An overview of the Riemannian metrics on spaces of curves using the {H}amiltonian approach, Appl. Comput. Harmon. Anal., 23 (2007), 74-113.  doi: 10.1016/j.acha.2006.07.004.

[23]

W. Mio and A. Srivastava, Elastic-string models for representation and analysis of planar shapes, In Computer Vision and Pattern Recognition, 2004. CVPR 2004. Proceedings of the 2004 IEEE Computer Society Conference on, volume 2, pages (2004), 10–15. doi: 10.1109/CVPR.2004.1315138.

[24]

W. MioA. Srivastava and S. Joshi, On shape of plane elastic curves, Int. J. Comput. Vision, 73 (2007), 307-324.  doi: 10.1007/s11263-006-9968-0.

[25]

T. Pejsa and I. Pandzic, State of the art in example-based motion synthesis for virtual characters in interactive applications, Computer Graphics Forum, 29 (2010), 202-226.  doi: 10.1111/j.1467-8659.2009.01591.x.

[26]

C. SamirP.-A. AbsilA. Srivastava and E. Klassen, A gradient-descent method for curve fitting on Riemannian manifolds, Found. Comput. Math., 12 (2012), 49-73.  doi: 10.1007/s10208-011-9091-7.

[27]

J. Shah, $H^0$-$ type Riemannian metrics on the space of planar curves, Quart. Appl. Math., 66 (2008), 123-137.  doi: 10.1090/S0033-569X-07-01084-4.

[28]

J. Shah, An $H^2$ Riemannian metric on the space of planar curves modulo similitudes, Adv. in Appl. Math., 51 (2013), 483-506.  doi: 10.1016/j.aam.2013.06.003.

[29]

E. Sharon and D. Mumford, 2D-shape analysis using conformal mapping, International Journal of Computer Vision, 70 (2004), 55-75.  doi: 10.1109/CVPR.2004.1315185.

[30]

A. SrivastavaE. KlassenS. Joshi and I. Jermyn, Shape analysis of elastic curves in euclidean spaces, Pattern Analysis and Machine Intelligence, IEEE Transactions on, 33 (2011), 1415-1428.  doi: 10.1109/TPAMI.2010.184.

[31]

J. SuS. KurtekE. Klassen and A. Srivastava, Statistical analysis of trajectories on Riemannian manifolds: bird migration, hurricane tracking and video surveillance, Ann. Appl. Stat., 8 (2014), 530-552.  doi: 10.1214/13-AOAS701.

[32]

Q. XieS. KurtekE. Klassen and A. Srivastava, Analysis of AneuRisk65 data: Elastic shape registration of curves, Electron. J. Stat., 8 (2014), 1920-1929.  doi: 10.1214/14-EJS938D.

[33]

L. Younes, Computable elastic distances between shapes, SIAM J. Appl. Math., 58 (1998), 565-586 (electronic).  doi: 10.1137/S0036139995287685.

[34]

L. YounesP. W. MichorJ. Shah and D. Mumford, A metric on shape space with explicit geodesics, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 19 (2008), 25-57.  doi: 10.4171/RLM/506.

Figure 1.  The dynamic programming algorithm can be sped up massively by only considering predecessor indices $(k, l)$ close to the current index $(i, j)$ in (29)
Figure 2.  This skeleton, which is based on data from the CMU Graphics Lab Motion Capture Database [12], was used for the animation experiments. Figure taken from [14]
Figure 3.  Effect of picking different feature points when matching two shapes. The top left figure shows results for shape matching using only an elastic energy functional without feature points. The remaining figures show matching results for different combinations of feature points. Corresponding markers on the left and right are matched, resulting in different paths between the given curves
Figure 4.  Effect of picking different feature points when matching two different hand shapes. In the top row, no feature points were set. The purely elastic matching produces distorted shapes along the geodesic path between the two hand shapes. In the middle row, feature points were set to match the tips of ring and index fingers correspondingly. This results in more natural interpolated shapes. In the bottom row, we see how incorrect feature matches cause some fingers to merge and new fingers to grow along the interpolation between the two shapes. Corresponding markers on the left and right are matched, resulting in different paths between the given curves. Colors along the curves indicate parametrization
Figure 5.  Example of using various methods to interpolate between two walking animations. The blue and orange lines are the trajectories of the left and right feet respectively. Note in particular how the two walking animations have different numbers of steps and how the various interpolated animations struggle with that. We have from left to right and top to bottom the following methods: linear interpolation of the Euler angles, elastic matching without reparametrization, elastic matching with reparametrization and finally elastic and feature point matching with reparametrization
Figure 6.  Example of using various methods to interpolate between two walking animations stepping over an obstacle. The blue and orange lines are the trajectories of the left and right feet respectively. We have from left to right and top to bottom the following methods: linear interpolation of the Euler angles, elastic matching without reparametrization, elastic matching with reparametrization and finally elastic and feature point matching with reparametrization
Table 1.  Computation times for Fig. 4
No. of pointsElastic reparam.Correct featuresIncorrect features
10017 s (6)11 s (3)22 s (6)
20032 s (6)22 s (3)50 s (6)
40067 s (6)48 s (3)95 s (8)
No. of pointsElastic reparam.Correct featuresIncorrect features
10017 s (6)11 s (3)22 s (6)
20032 s (6)22 s (3)50 s (6)
40067 s (6)48 s (3)95 s (8)
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